Math 515 homework #10 Due date: 19980223 D. J. Bernstein 19980216 Exercise 23. Let R be a ring. Let M be an R-module. Let t be an integer. Assume that len_{R} L ≤ t for every finitely generated R-submodule L of M. Prove that len_{R} M ≤ t.

Local rings. Let R be a ring. Let P be the set of non-units of R. Every proper ideal of R is contained in P. R is local if P is an ideal of R; then P is the unique maximal ideal of R. For example, every field is a local ring. Exercise 24. Let R be a finite ring. Prove that the number of units in R is #R ∏_{P ∈Spec R}(1 − #P/#R). (Hint: First consider the case when R is local.) Exercise 25. Let R be an Artin ring. Let u be an element of R. Prove that there is a unit v and an integer n for which u^{n+1} = vu^{n}. (In particular, if u is cancellable, then it is a unit.) Exercise 26. Let R be an Artin ring. Let U be a multiplicative subset of R. Prove that the natural ring homomorphism from R to U^{−1}R is surjective. (This generalizes the fact that any Artin domain is a field.)